The given mathematical expression is a division of fractions involving polynomials. Let's break it down step by step.

The original expression is:

$\frac{6y}{y^2 - 36} : \left( \frac{y^2 + 12y + 36}{y^2 - 6y} : \frac{y + 6}{3y^{-2}} \right)$

Step 1: Factor the denominators and numerators

• For the first fraction: $y^2 - 36 = (y - 6)(y + 6)$ So, the first fraction becomes: $\frac{6y}{(y - 6)(y + 6)}$

• For the second fraction inside the parentheses: $y^2 + 12y + 36 = (y + 6)(y + 6)$ So the second fraction becomes: $\frac{(y + 6)(y + 6)}{y^2 - 6y}$ We can factor $y^2 - 6y$ as: $y^2 - 6y = y(y - 6)$ So, the second fraction becomes: $\frac{(y + 6)(y + 6)}{y(y - 6)}$

• For the third fraction: $\frac{y + 6}{3y^{-2}} = \frac{y + 6}{\frac{3}{y^2}} = \frac{(y + 6)y^2}{3}$

Step 2: Simplify the division operation

The expression is now: $\frac{6y}{(y - 6)(y + 6)} : \left( \frac{(y + 6)(y + 6)}{y(y - 6)} : \frac{(y + 6)y^2}{3} \right)$

We will now handle the innermost division first: $\frac{(y + 6)(y + 6)}{y(y - 6)} : \frac{(y + 6)y^2}{3}$

Dividing by a fraction is the same as multiplying by its reciprocal: $= \frac{(y + 6)(y + 6)}{y(y - 6)} \times \frac{3}{(y + 6)y^2}$

Now, simplify:

• One factor of $y + 6$ cancels out.
• The remaining expression is: $= \frac{3(y + 6)}{y(y - 6)y^2}$

This simplifies to: $\frac{3(y + 6)}{y^3(y - 6)}$

Step 3: Perform the final division

Now, return to the original expression: $\frac{6y}{(y - 6)(y + 6)} : \frac{3(y + 6)}{y^3(y - 6)}$ This is the same as multiplying by the reciprocal of the second fraction: $= \frac{6y}{(y - 6)(y + 6)} \times \frac{y^3(y - 6)}{3(y + 6)}$

Now simplify:

• $y - 6$ cancels.
• One $y + 6$ cancels.
• One factor of $y$ cancels.

The remaining expression is: $= \frac{6y^2}{3} = 2y^2$

Thus, the simplified expression is: $2y^2$

Would you like more details or have any questions about this solution?

Related Questions:

1. How do you factor quadratic expressions like $y^2 - 36$?
2. What are the rules for dividing and multiplying fractions involving polynomials?
3. How can you simplify complex algebraic fractions using cancellation?
4. What is the significance of factoring in simplifying rational expressions?
5. How does multiplying by the reciprocal simplify division of fractions?

Tip:

When dividing fractions, remember to multiply by the reciprocal of the second fraction to avoid confusion.